Subgroups — #28481872

Order 15104.

408 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433052	C₂	1	1	yes	maximal, characteristic
2	C₄×#14273662	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422851	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422536	C₂	1	1	yes	maximal, characteristic
5	#21422533	C₂	1	1	yes	maximal, characteristic
6	#21421987	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1299	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11745685	C₂²	1	1	yes	characteristic
9	#16837526	C₄	1	1	yes	characteristic
10	#16837526	C₄	1	1	yes	characteristic
11	#16837467	C₄	1	1	yes	characteristic
12	#16837467	C₄	1	1	yes	characteristic
13	(C₂²⋊D₅₉)⋊C₈	—	2	1	no	—
14	(C₂²⋊D₅₉)⋊C₈	—	2	1	no	—
15	#16837453	—	2	1	no	—
16	C₂×#14273662	—	2	1	no	—
17	#16837426	C₂²	1	1	yes	characteristic
18	C₄×#11745664	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₈×D₅₉	—	2	1	no	—
21	#16837166	—	2	1	no	—
22	#16837152	—	2	1	no	—
23	#16837152	—	2	1	no	—
24	#16837134	—	2	1	no	—
25	#16837132	—	2	1	no	—
26	#16837131	—	2	1	no	—
27	#16837129	—	2	1	no	—
28	C₅₉×#431	C₂²	1	1	yes	characteristic
29	C₅₉×#423	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×#11745685	Q₈	1	1	yes	characteristic
32	C₂×#11745664	—	2	1	no	—
33	C₂×#11745664	—	2	1	no	—
34	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
36	C₂×C₈×D₅₉	—	2	1	no	—
37	C₂×C₈×D₅₉	—	2	1	no	—
38	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
39	C₄×Dic₁₁₈	D₄	1	1	yes	characteristic
40	C₄×D₂₃₆	C₂×C₄	1	1	yes	characteristic
41	C₄×D₂₃₆	D₄	1	1	yes	characteristic
42	C₄×C₂²⋊D₅₉	—	2	1	no	—
43	C₄×C₂²⋊D₅₉	—	2	1	no	—
44	#14273686	Q₈	1	1	yes	characteristic
45	#14273684	—	2	1	no	—
46	C₄²×D₅₉	—	2	1	no	—
47	#14273662	—	2	4	no	—
48	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
49	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
50	Dic₅₉⋊C₈	—	2	1	no	—
51	Dic₅₉⋊C₈	—	2	1	no	—
52	(C₂×D₅₉)⋊C₈	—	2	1	no	—
53	(C₂×D₅₉)⋊C₈	—	2	1	no	—
54	#14273638	—	2	1	no	—
55	C₄×C₅₉⋊C₈	—	2	1	no	—
56	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
59	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
60	C₈×Dic₅₉	—	2	1	no	—
61	C₈×Dic₅₉	—	2	1	no	—
62	C₂×M₁₆×C₅₉	—	2	1	no	—
63	C₂×M₁₆×C₅₉	—	2	1	no	—
64	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₅₉	—	2	1	no	—
66	C₄⋊C₈×C₅₉	—	2	1	no	—
67	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
68	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊C₈×C₅₉	—	2	1	no	—
70	#97×C₅₉	—	2	1	no	—
71	C₄×C₈×C₅₉	—	2	1	no	—
72	C₄×C₈×C₅₉	—	2	1	no	—
73	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
76	C₂×C₂²⋊D₅₉	—	2	1	no	—
77	#11745685	—	2	2	no	—
78	#11745685	—	2	2	no	—
79	C₂×D₂₃₆	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₅₉	—	2	1	no	—
81	C₂×C₄×D₅₉	—	2	1	no	—
82	C₂×C₄×D₅₉	—	2	1	no	—
83	C₂×Dic₁₁₈	C₄⋊C₄	1	1	yes	characteristic
84	M₁₆×C₅₉	—	2	2	no	—
85	M₁₆×C₅₉	—	4	1	no	—
86	M₁₆×C₅₉	—	8	1	no	—
87	C₂×C₈×C₅₉	—	2	1	no	—
88	C₂×C₈×C₅₉	—	2	1	no	—
89	C₂×C₈×C₅₉	—	4	1	no	—
90	C₂×C₈×C₅₉	—	4	1	no	—
91	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₉	#41	1	1	yes	characteristic
95	C₂²⋊Dic₅₉	—	2	1	no	—
96	(C₂×D₅₉)⋊C₄	—	2	1	no	—
97	(C₂×D₅₉)⋊C₄	—	2	1	no	—
98	C₄⋊Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
99	C₄⋊Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
100	Dic₅₉⋊C₄	—	2	1	no	—
101	Dic₅₉⋊C₄	—	2	1	no	—
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₄×Dic₅₉	—	2	1	no	—
105	#11745664	—	2	2	no	—
106	#11745664	—	4	1	no	—
107	#11745664	—	8	1	no	—
108	C₂×C₅₉⋊C₈	—	2	2	no	—
109	C₂×C₅₉⋊C₈	—	4	1	no	—
110	C₂×C₅₉⋊C₈	—	4	1	no	—
111	C₂×C₅₉⋊C₈	—	4	1	no	—
112	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
116	C₈×D₅₉	—	4	2	no	—
117	C₈×D₅₉	—	4	2	no	—
118	C₂³×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂²×D₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	—	2	1	no	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	—	2	1	no	—
125	C₂×C₄×C₅₉	—	2	1	no	—
126	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
129	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂²⋊D₅₉	—	2	4	no	—
133	C₂×Dic₅₉	—	2	1	no	—
134	C₂×Dic₅₉	—	2	1	no	—
135	C₂×Dic₅₉	—	2	1	no	—
136	D₂₃₆	—	2	1	no	—
137	D₂₃₆	#108	1	2	yes	complemented
138	C₄×D₅₉	—	2	4	no	—
139	C₄×D₅₉	—	4	1	no	—
140	C₄×D₅₉	—	4	1	no	—
141	C₄×D₅₉	—	4	1	no	—
142	C₄×D₅₉	—	4	1	no	—
143	Dic₁₁₈	—	2	1	no	—
144	Dic₁₁₈	#108	1	2	yes	complemented
145	C₈×C₅₉	—	4	2	no	—
146	C₈×C₅₉	—	4	1	no	—
147	C₈×C₅₉	—	4	1	no	—
148	C₈×C₅₉	—	8	1	no	—
149	C₅₉⋊C₈	—	2	1	no	—
150	C₅₉⋊C₈	—	2	1	no	—
151	C₅₉⋊C₈	—	2	1	no	—
152	C₅₉⋊C₈	—	4	2	no	—
153	C₅₉⋊C₈	—	8	1	no	—
154	C₅₉⋊C₈	#104	1	2	yes	—
155	#8670	—	59	1	no	maximal, Sylow
156	C₂²×C₅₉	—	2	1	no	—
157	C₂²×C₅₉	—	2	1	no	—
158	C₂²×C₅₉	—	2	1	no	—
159	C₂²×C₅₉	#380	1	1	yes	characteristic
160	C₂×D₅₉	—	2	2	no	—
161	C₂×D₅₉	—	4	1	no	—
162	C₂×D₅₉	—	4	1	no	—
163	C₄×C₅₉	—	2	1	no	—
164	C₄×C₅₉	—	2	1	no	—
165	C₄×C₅₉	—	2	1	no	—
166	C₄×C₅₉	C₂×#104	1	1	yes	characteristic
167	C₄×C₅₉	#421	1	1	yes	characteristic
168	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
169	C₄×C₅₉	C₂×#108	1	1	yes	characteristic
170	C₄×C₅₉	#471	1	1	yes	characteristic
171	C₄×C₅₉	#471	1	1	yes	characteristic
172	Dic₅₉	—	2	2	no	—
173	Dic₅₉	—	4	1	no	—
174	Dic₅₉	—	4	1	no	—
175	C₈×#41	—	59	1	no	—
176	C₄×#108	—	59	1	no	—
177	#1566	—	59	1	no	—
178	#1489	—	59	1	no	—
179	#1489	—	59	1	no	—
180	#1299	—	59	1	no	—
181	#1299	D₅₉	1	1	yes	characteristic, complemented
182	C₂×C₅₉	—	4	1	no	—
183	C₂×C₅₉	#1806	1	1	yes	characteristic
184	C₂×C₅₉	#1810	1	1	yes	characteristic
185	C₂×C₅₉	#1815	1	1	yes	characteristic
186	D₅₉	—	4	2	no	—
187	C₄×#41	—	59	1	no	—
188	C₈×Q₈	—	59	1	no	—
189	C₈×Q₈	—	59	1	no	—
190	C₈×D₄	—	59	1	no	—
191	C₈×D₄	—	59	1	no	—
192	C₈×D₄	—	118	1	no	—
193	C₈×D₄	—	118	1	no	—
194	#431	—	118	1	no	—
195	#431	C₂×D₅₉	1	1	yes	characteristic
196	C₂×#108	—	118	1	no	—
197	#423	—	59	1	no	—
198	#423	C₂×D₅₉	1	1	yes	characteristic
199	C₄×M₁₆	—	59	1	no	—
200	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
201	C₂×C₄×C₈	—	59	1	no	—
202	C₂×C₄×C₈	—	118	1	no	—
203	#341	—	118	1	no	—
204	#334	—	118	1	no	—
205	#334	—	118	1	no	—
206	Q₈⋊C₈	—	118	1	no	—
207	Q₈⋊C₈	—	118	1	no	—
208	D₄⋊C₈	—	118	1	no	—
209	D₄⋊C₈	—	118	1	no	—
210	C₅₉	#8670	1	1	yes	characteristic, complemented, Sylow
211	C₂×#41	—	59	1	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	118	1	no	—
215	C₂×M₁₆	—	118	1	no	—
216	C₂²×C₈	—	59	1	no	—
217	C₂²×C₈	—	59	1	no	—
218	C₂²×C₈	—	118	1	no	—
219	C₂²×C₈	—	118	1	no	—
220	C₄×Q₈	—	59	1	no	—
221	C₄×Q₈	—	59	1	no	—
222	C₄×D₄	—	59	1	no	—
223	C₄×D₄	—	59	1	no	—
224	C₄×D₄	—	118	1	no	—
225	C₄×D₄	—	118	1	no	—
226	#117	—	59	1	no	—
227	#117	—	118	1	no	—
228	C₂×C₄²	—	118	1	no	—
229	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
230	#108	—	118	4	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	—	118	1	no	—
234	C₄⋊C₈	—	118	1	no	—
235	C₄⋊C₈	—	118	1	no	—
236	C₄⋊C₈	—	118	1	no	—
237	C₄⋊C₈	C₄×D₅₉	1	1	yes	characteristic
238	C₄⋊C₈	C₄×D₅₉	1	1	yes	characteristic
239	C₂²⋊C₈	—	2	1	no	—
240	C₂²⋊C₈	—	118	1	no	—
241	C₂²⋊C₈	—	118	1	no	—
242	#97	—	2	1	no	—
243	#97	—	118	1	no	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	—	2	1	no	—
246	C₄×C₈	—	59	1	no	—
247	C₄×C₈	—	59	1	no	—
248	C₄×C₈	—	59	1	no	—
249	C₄×C₈	—	59	1	no	—
250	C₄×C₈	—	118	1	no	—
251	C₄×C₈	—	118	1	no	—
252	C₄×C₈	—	118	1	no	—
253	#41	—	118	2	no	—
254	#41	—	118	2	no	—
255	C₂×Q₈	—	59	1	no	—
256	C₂×D₄	—	59	1	no	—
257	C₂×D₄	—	118	1	no	—
258	C₂²×C₄	—	118	1	no	—
259	C₂²×C₄	—	118	1	no	—
260	C₂²×C₄	—	118	1	no	—
261	C₂²×C₄	#11745685	1	1	yes	characteristic
262	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
263	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	8	1	no	—
267	M₁₆	—	118	2	no	—
268	M₁₆	—	472	1	no	—
269	M₁₆	—	236	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	59	1	no	—
275	C₂×C₈	—	59	1	no	—
276	C₂×C₈	—	59	1	no	—
277	C₂×C₈	—	59	1	no	—
278	C₂×C₈	—	118	2	no	—
279	C₂×C₈	—	236	2	no	—
280	C₂×C₈	—	236	2	no	—
281	C₂×C₈	—	236	1	no	—
282	C₂×C₈	—	236	1	no	—
283	C₂×C₈	—	236	1	no	—
284	C₄⋊C₄	—	59	1	no	—
285	C₄⋊C₄	—	59	1	no	—
286	C₄⋊C₄	—	118	1	no	—
287	C₄⋊C₄	—	118	1	no	—
288	C₂²⋊C₄	—	118	1	no	—
289	C₂²⋊C₄	—	118	1	no	—
290	C₂²⋊C₄	—	118	1	no	—
291	C₄²	—	118	1	no	—
292	C₄²	—	118	1	no	—
293	C₄²	—	118	1	no	—
294	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
295	C₄²	#11745685	1	1	yes	characteristic
296	C₄²	D₄×D₅₉	1	1	yes	characteristic
297	C₄²	D₄×D₅₉	1	1	yes	characteristic
298	C₂³	—	118	1	no	—
299	C₂³	#14273705	1	1	yes	characteristic
300	Q₈	—	59	2	no	—
301	Q₈	—	118	1	no	—
302	D₄	—	59	2	no	—
303	D₄	—	118	4	no	—
304	D₄	—	118	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	118	4	no	—
312	C₂×C₄	—	118	1	no	—
313	C₂×C₄	—	118	1	no	—
314	C₂×C₄	—	118	1	no	—
315	C₂×C₄	—	236	1	no	—
316	C₂×C₄	—	236	1	no	—
317	C₂×C₄	—	236	1	no	—
318	C₂×C₄	—	236	1	no	—
319	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
320	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
321	C₂×C₄	#14273684	1	1	yes	characteristic
322	C₂×C₄	#14273700	1	1	yes	characteristic
323	C₂×C₄	#14273700	1	1	yes	characteristic
324	C₂×C₄	#14273705	1	1	yes	characteristic
325	C₈	—	4	2	no	—
326	C₈	—	4	1	no	—
327	C₈	—	4	1	no	—
328	C₈	—	8	1	no	—
329	C₈	—	59	2	no	—
330	C₈	—	118	1	no	—
331	C₈	—	118	1	no	—
332	C₈	—	118	1	no	—
333	C₈	—	472	1	no	—
334	C₈	—	236	2	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	118	2	no	—
339	C₂²	—	236	1	no	—
340	C₂²	—	236	1	no	—
341	C₂²	#16837261	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	118	2	no	—
346	C₄	—	236	1	no	—
347	C₄	—	236	1	no	—
348	C₄	#104×D₅₉	1	1	yes	characteristic
349	C₄	#16837417	1	1	yes	characteristic
350	C₄	#108×D₅₉	1	1	yes	characteristic
351	C₄	#108×D₅₉	1	1	yes	characteristic
352	C₄	#16837664	1	1	yes	characteristic
353	C₄	#16837668	1	1	yes	characteristic
354	C₂	—	4	1	no	—
355	C₂	—	236	2	no	—
356	C₂	#21424115	1	1	yes	characteristic
357	C₂	#21424133	1	1	yes	characteristic
358	C₂	#21424154	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273700
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1299	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂×#11745685	Q₈
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1299	D₅₉
hypocenter	C₅₉	#8670
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1299

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄×#14273662	C₂	semidirect
3. #21422851	C₂	semidirect
6. #21421987	C₂	semidirect
7. C₅₉×#1299	C₂	semidirect
137. D₂₃₆	#108	semidirect
144. Dic₁₁₈	#108	semidirect
181. #1299	D₅₉	semidirect
210. C₅₉	#8670	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21433052	2
C₄×#14273662	2
#21422851	2
#21422536	2
#21422533	2
#21421987	2
C₅₉×#1299	2
#8670	59